Curves Of Genus Research Articles

On the Rank of Quadratic Equations for Curves of High Degree

Let $${\mathcal {C}} \subset {{\mathbb {P}}}^r$$ be a linearly normal curve of arithmetic genus g and degree d. In Saint-Donat (CR Acad Sci Paris Ser A 274: 324–327, 1972), B. Saint-Donat proved that the homogeneous ideal $$I({\mathcal {C}})$$ of $${\mathcal {C}}$$ is generated by quadratic equations of rank at most 4 whenever $$d \ge 2g+2$$ . Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that $$I({\mathcal {C}})$$ admits a determinantal presentation if $$d \ge 4g+2$$ . In this paper, we will show that $$I({\mathcal {C}})$$ can be generated by quadratic equations of rank 3 if either $$g=0,1$$ and $$d \ge 2g+2$$ or else $$g \ge 2$$ and $$d \ge 4g+4$$ .

Moments of quadratic Dirichlet L-functions over function fields

In this paper, we establish the expected order of magnitude of the kth-moment of quadratic Dirichlet L-functions associated to hyperelliptic curves of genus g as well as of prime moduli over a fixed finite field Fq for all real k≥0.

Perverse sheaves on $\mathbf{C}^{2}$ without vanishing cycles at the origin along a general plane curve with singularities

Generalizing MacPherson-Vilonen’s method [2] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by Neto and Silva [5]. The main result is an equivalence of categories between the category of perverse sheaves on $ \mathbf{C}^{2}$ stratified with respect to a singular plane curve and the category of $n$-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case.

The modularity of elliptic curves over all but finitely many totally real fields of degree 5

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.

A note on semiorthogonal indecomposability of some Cohen-Macaulay varieties

In this short note, we observe that the criterion proven in [12] for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes.

SN 2019zrk, a bright SN 2009ip analog with a precursor

We present photometric and spectroscopic observations of the Type IIn supernova SN 2019zrk (also known as ZTF 20aacbyec). The SN shows a > 100 day precursor, with a slow rise, followed by a rapid rise toM ≈ −19.2 in therandgbands. The post-peak light-curve decline is well fit with an exponential decay with a timescale of ∼39 days, but it shows prominent undulations, with an amplitude of ∼1 mag. Both the light curve and spectra are dominated by an interaction with a dense circumstellar medium (CSM), probably from previous mass ejections. The spectra evolve from a scattering-dominated Type IIn spectrum to a spectrum with strong P-Cygni absorptions. The expansion velocity is high, ∼16 000 km s−1, even in the last spectra. The last spectrum ∼110 days after the main eruption reveals no evidence for advanced nucleosynthesis. From analysis of the spectra and light curves, we estimate the mass-loss rate to be ∼4 × 10−2 M⊙yr−1for a CSM velocity of 100 km s−1, and a CSM mass of 1 M⊙. We find strong similarities for both the precursor, general light curve, and spectral evolution with SN 2009ip and similar SNe, although SN 2019zrk displays a brighter peak magnitude. Different scenarios for the nature of the 09ip-class of SNe, based on pulsational pair instability eruptions, wave heating, and mergers, are discussed.

Équation de Pell–Abel et applications

In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r>g. This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive k-differential on an hyperelliptic curve of genus g with a unique zero of order k(2g-2) for every (k,g)≠(2,2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n>2g.

Vulnerability Assessment of School Buildings: Generic Versus Building-Specific Fragility Curves

ABSTRACT The aim of this paper is to investigate whether existing generic fragility curves for various residential building typologies can be safely applied for the vulnerability assessment of school buildings. We select three representative typologies of actual reinforced concrete (RC) school buildings in Thessaloniki, Greece and we conduct a detailed fragility analysis combining field inspection and structural monitoring data to advanced numerical modeling. Building-specific fragility and vulnerability curves are derived for the three school buildings based on 3D incremental dynamic analysis (IDA) of the calibrated numerical models. The comparisons of the developed curves with generic ones showed their significant differences.

Algebraic Integrability of Planar Polynomial Vector Fields by Extension to Hirzebruch Surfaces

We study algebraic integrability of complex planar polynomial vector fields \(X=A (x,y)(\partial /\partial x) + B(x,y) (\partial /\partial y) \) through extensions to Hirzebruch surfaces. Using these extensions, each vector field X determines two infinite families of planar vector fields that depend on a natural parameter which, when X has a rational first integral, satisfy strong properties about the dicriticity of the points at the line \(x=0\) and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if X has a rational first integral, we provide a region in \({\mathbb {R}}_{\ge 0}^2\) that contains all the pairs (i, j) corresponding to monomials \(x^i y^j\) involved in the generic invariant curve of X.

The Influence of Listeners’ Mood on Equalization-Based Listening Experience

Using equalization to improve sound listening experience is a well-established topic among the audio society. Finding a general equalization curve is a difficult task because of spectral content influenced by the reproduction system (loudspeakers and room environment) and personal preference diversity. Listeners’ mood is said to be a factor that affects the individual equalization preference. In this study, the effect of a listener’s mood on equalization preference is tried to be investigated. Starting from an experiment with fifty-two listeners, considering five predefined equalization curves and a database of ten music excerpts, the relationship between listeners’ mood and preferred sound equalization has been studied. The main findings of this study showed that the “High-frequency boosting” equalization was the most preferred among participants. However, the “High-frequency boosting” preference of low-aroused people was slightly lower than the high aroused listeners, increasing the preference of the “Low-frequency boosting”.

Schottky spaces and universal Mumford curves over $${\mathbb {Z}}$$

For every integer \(g \ge 1\) we define a universal Mumford curve of genus g in the framework of Berkovich spaces over \({\mathbb {Z}}\). This is achieved in two steps: first, we build an analytic space \({{\mathcal {S}}}_g\) that parametrizes marked Schottky groups over all valued fields. We show that \({{\mathcal {S}}}_g\) is an open, connected analytic space over \({\mathbb {Z}}\). Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of \({{\mathcal {S}}}_g\), both locally and globally. As a result, we can define the universal Mumford curve \({{\mathcal {C}}}_g\) as a relative curve over \({{\mathcal {S}}}_g\) such that every Schottky uniformized curve can be described as a fiber of a point in \({{\mathcal {S}}}_g\). We prove that the curve \({{\mathcal {C}}}_g\) is itself uniformized by a universal Schottky group acting on the relative projective line \(\mathbb {P}^1_{{{\mathcal {S}}}_g}\). Finally, we study the action of the group \(\hbox {Out}(F_g)\) of outer automorphisms of the free group with g generators on \({{\mathcal {S}}}_g\), describing the quotient \(\hbox {Out}(F_g) \backslash {{\mathcal {S}}}_g\) in the archimedean and non-archimedean cases. We apply this result to compare the non-archimedean Schottky space with constructions arising from geometric group theory and the theory of moduli spaces of tropical curves.

Limit of Weierstrass measure on stable curves

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g \geqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne–Mumford compactification $\overline{\mathcal{M}}_g$ of the moduli $\mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $\mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.

Algebraic geometric secret sharing schemes over large fields are asymptotically threshold

In Chen-Cramer Crypto 2006 paper \cite{cc} algebraic geometric secret sharing schemes were proposed such that the Fundamental Theorem in Information-Theoretically Secure Multiparty Computation by Ben-Or, Goldwasser and Wigderson \cite{BGW88} and Chaum, Crepeau and Damgard \cite{CCD88} can be established over constant-size base finite fields. These algebraic geometric secret sharing schemes defined by a curve of genus $g$ over a constant size finite field ${\bf F}_q$ is quasi-threshold in the following sense, any subset of $u \leq T-1$ players (non qualified) has no information of the secret and any subset of $u \geq T+2g$ players (qualified) can reconstruct the secret. It is natural to ask that how far from the threshold these quasi-threshold secret sharing schemes are? How many subsets of $u \in [T, T+2g-1]$ players can recover the secret or have no information of the secret? In this paper it is proved that almost all subsets of $u \in [T,T+g-1]$ players have no information of the secret and almost all subsets of $u \in [T+g,T+2g-1]$ players can reconstruct the secret when the size $q$ goes to the infinity and the genus satisfies $\lim \frac{g}{\sqrt{q}}=0$. Then algebraic geometric secret sharing schemes over large finite fields are asymptotically threshold in this case. We also analyze the case when the size $q$ of the base field is fixed and the genus goes to the infinity.

Smooth tropical complete intersection curves of genus 3 in $$\mathbb {R}^3$$

We develop a method for describing the tropical complete intersection of a tropical hypersurface and a tropical plane in \(\mathbb {R}^3\). This involves a method for determining the topological type of the intersection of a tropical plane curve and \(\mathbb {R}_{\le 0}^2\) by using a polyhedral complex. As an application, we study smooth tropical complete intersection curves of genus 3 in \(\mathbb {R}^3\). In particular, we show that there are no smooth tropical complete intersection curves in \(\mathbb {R}^3\) whose skeletons are the lollipop graph of genus 3. This gives a partial answer to a problem of Morrison in [6].

Symplectic geometry of p-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles

The aim of the present paper is to provide a new aspect of the $p$-adic Teichm\"{u}ller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichm\"{u}ller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ classifying $p$-adic canonical liftings of ordinary nilpotent indigenous bundles. The formal stack $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ embodies a $p$-adic analogue of uniformization of hyperbolic Riemann surfaces, as well as a hyperbolic analogue of Serre-Tate theory of ordinary abelian varieties. In the present paper, the canonical symplectic structure on the cotangent bundle $T^\vee_{\mathbb{Z}_p} \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ of $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ is compared to Goldman's symplectic structure defined on $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ after base-change by the projection $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}} \rightarrow \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$. We can think of this comparison as a $p$-adic analogue of certain results in the theory of projective structures on Riemann surfaces proved by S. Kawai and other mathematicians.

Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication

We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods to find CM fields K K for which there exist both hyperelliptic and non-hyperelliptic curves whose Jacobian has complex multiplication by Z K \mathbb {Z}_K . More precisely, we find all sextic CM fields K K in the LMFDB for which (heuristically) Jacobians of both types with CM by Z K \mathbb {Z}_K exist. There turn out to be 14 such fields among the 547,156 sextic CM fields that the LMFDB contains. We determine invariants of the corresponding curves, and in the simplest case we also give an explicit defining equation.

Explicit coverings of families of elliptic surfaces by squares of curves

We show that, for each $$n>0$$ , there is a family of elliptic surfaces which are covered by the square of a curve of genus $$2n+1$$ , and whose Hodge structures have an action by $${{\mathbb {Q}}}(\sqrt{-n})$$ . By considering the case $$n=3$$ , we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in $${\mathbb {P}}^4$$ with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga–Satake–Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.

The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

Constructing reducible Brill–Noether curves II

This paper is the second of a two-part series by the author devoted to the following fundamental problem in the theory of algebraic curves in projective space: Which reducible curves arise as limits of smooth curves of general moduli? Special cases of this question studied by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ), and others have been critical in the resolution of many problems in the theory of algebraic curves over the past half century. In this paper, we give sharp bounds on this problem for space curves, when the nodes are general points and the components are general in moduli. We also systematically study a variant in projective spaces of arbitrary dimension when the nodes are general in a hyperplane. The results given here significantly extend those cases established in the previous paper in this series (Eric Larson, Constructing reducible Brill-Noether curves, To appear in documentamathematica, arxiv:1603.02301), as well as those cases established by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ) , and others. As explained in (Eric Larson, Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980), the reducible curves constructed in this paper also play a critical role in the author’s proof of the maximal rank conjecture in a subsequent paper (Eric Larson Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980).

Stability of normal bundles of space curves

In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g \geq 2$ is stable if and only if $(d,g) \not\in \{ (5,2), (6,4) \}$. When $g\leq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree.